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  "body": "\n## Summary\n\nThe $n$-th Weyl algebra $A_n(k)$ over a field $k$ is the algebra generated by $x_1, \\dots, x_n$ and $y_1, \\dots, y_n$ with the relations $y_i x_i - x_i y_i = 1$.\nAll other pairs of generators commute.\nIn characteristic zero it is the [[ring of differential operators|GBWQF3HZG0]] with polynomial coefficients, where $x_i$ acts by multiplication and $y_i$ by $\\partial / \\partial x_i$.\nIn characteristic zero it is the basic example of a simple Noetherian domain that is not a division ring.\nIt also models the canonical commutation relation of quantum mechanics.\n\n## Explanations\n\n### L1 human\n\nTake polynomials in $x$ and the operator $D = d/dx$ that differentiates.\nBy the product rule, $D(x f) = x D(f) + f$, so the operators satisfy $D x - x D = 1$.\nThe first Weyl algebra is the algebra built from two symbols with exactly this rule.\nEvery element can be written uniquely as a finite sum $\\sum a_{ij} x^i D^j$, like a polynomial in two variables that do not commute.\n",
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